How can i bake an animated and extruded curve to a mesh animation?
The blue mesh isn't a real mesh it is generated from a curve so it can't be used in the Boolean modifier ( and other mesh dependent modifiers ) , also the text object( red one ) is a curve so it can't be used in the Boolean modifier.

You can convert  the curve into a mesh :
 - select the curve and hit Alt+C and choose Mesh from Curve 
Note: beware after converting to a mesh you loose the curve editing options ( including this animation )

The animation you have can be replicated using the build modifier  on the converted mesh :



now you can have both the animation and the Boolean modifier on this mesh